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In mathematics, a **tangle** is generally one of two related concepts:

- Tangle diagrams
- Rational and algebraic tangles
- Operations on tangles
- Conway notation
- Applications
- See also
- References
- Further reading
- External links

- In John Conway's definition, an
*n*-**tangle**is a proper embedding of the disjoint union of*n*arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2*n*marked points on the ball's boundary. - In link theory, a tangle is an embedding of
*n*arcs and*m*circles into – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.

(A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, * Journal of Combinatorial Theory * B 59 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.)

The balance of this article discusses Conway's sense of tangles; for the link theory sense, see that article.

Two *n*-tangles are considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball fixed. **Tangle theory** can be considered analogous to knot theory except instead of closed loops we use strings whose ends are nailed down. See also braid theory.

Without loss of generality, consider the marked points on the 3-ball boundary to lie on a great circle. The tangle can be arranged to be in general position with respect to the projection onto the flat disc bounded by the great circle. The projection then gives us a **tangle diagram**, where we make note of over and undercrossings as with knot diagrams.

Tangles often show up as tangle diagrams in knot or link diagrams and can be used as building blocks for link diagrams, e.g. pretzel links.

A **rational tangle** is a 2-tangle that is homeomorphic to the trivial 2-tangle by a map of pairs consisting of the 3-ball and two arcs. The four endpoints of the arcs on the boundary circle of a tangle diagram are usually referred as NE, NW, SW, SE, with the symbols referring to the compass directions.

An arbitrary tangle diagram of a rational tangle may look very complicated, but there is always a diagram of a particular simple form: start with a tangle diagram consisting of two horizontal (vertical) arcs; add a "twist", i.e. a single crossing by switching the NE and SE endpoints (SW and SE endpoints); continue by adding more twists using either the NE and SE endpoints or the SW and SE endpoints. One can suppose each twist does not change the diagram inside a disc containing previously created crossings.

We can describe such a diagram by considering the numbers given by consecutive twists around the same set of endpoints, e.g. (2, 1, -3) means start with two horizontal arcs, then 2 twists using NE/SE endpoints, then 1 twist using SW/SE endpoints, and then 3 twists using NE/SE endpoints but twisting in the opposite direction from before. The list begins with 0 if you start with two vertical arcs. The diagram with two horizontal arcs is then (0), but we assign (0, 0) to the diagram with vertical arcs. A convention is needed to describe a "positive" or "negative" twist. Often, "rational tangle" refers to a list of numbers representing a simple diagram as described.

The **fraction** of a rational tangle is then defined as the number given by the continued fraction . The fraction given by (0,0) is defined as . Conway proved that the fraction is well-defined and completely determines the rational tangle up to tangle equivalence.^{ [1] } An accessible proof of this fact is given in:.^{ [2] } Conway also defined a fraction of an arbitrary tangle by using the Alexander polynomial.

There is an "arithmetic" of tangles with addition, multiplication, and reciprocal operations. An algebraic tangle is obtained from the addition and multiplication of rational tangles.

The **numerator closure** of a rational tangle is defined as the link obtained by joining the "north" endpoints together and the "south" endpoints also together. The **denominator closure** is defined similarly by grouping the "east" and "west" endpoints. Rational links are defined to be such closures of rational tangles.

One motivation for Conway's study of tangles was to provide a notation for knots more systematic than the traditional enumeration found in tables.

Tangles have been shown to be useful in studying DNA topology. The action of a given enzyme can be analysed with the help of tangle theory.^{ [3] }

In the mathematical theory of knots, the unknot, or **trivial knot**, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic to a geometrically round circle, the **standard unknot**.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In topology, **knot theory** is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, **R**^{3}. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

In topology, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, an **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

**Skein relations** are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invariants of the knot. If two diagrams have different polynomials, they represent different knots. In general, the converse does not hold.

In mathematics, a **knot** is an embedding of a circle *S*^{1} in 3-dimensional Euclidean space, **R**^{3}, considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term *knot* is also applied to embeddings of *S*^{ j} in *S*^{n}, especially in the case *j* = *n* − 2. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.

In knot theory, a branch of mathematics, the **trefoil knot** is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.

In mathematics, the **linking number** is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

In mathematics, specifically in topology, the operation of **connected sum** is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.

In mathematics, **geometric topology** is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

In knot theory, there are several competing notions of the quantity *writhe*, or *Wr*. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot in three-dimensional space and assumes real numbers as values. In both cases, *writhe* is a geometric quantity, meaning that while deforming a curve in such a way that does not change its topology, one may still change its *writhe*.

In mathematics, the **Kirby calculus** in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the **Kirby moves**. Using four-dimensional Cerf theory, he proved that if *M* and *N* are 3-manifolds, resulting from Dehn surgery on framed links *L* and *J* respectively, then they are homeomorphic if and only if *L* and *J* are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.

In the mathematical field of knot theory, the **Jones polynomial** is a knot polynomial discovered by Vaughan Jones in 1984.,. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.

In mathematical knot theory, a **link** is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a *trivial* reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an *n*-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension *n*. In this more precise terminology, a manifold is referred to as an ** n-manifold**.

In the mathematical field of knot theory, a **mutation** is an operation on a knot that can produce different knots. Suppose *K* is a knot given in the form of a knot diagram. Consider a disc *D* in the projection plane of the diagram whose boundary circle intersects *K* exactly four times. We may suppose that the disc is geometrically round and the four points of intersection on its boundary with *K* are equally spaced. The part of the knot inside the disc is a tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a **mutant** of *K*.

In knot theory, **Conway notation**, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it.

The concept of **alternating planar algebras** first appeared in the work of Hernando Burgos-Soto on the Jones polynomial of alternating tangles. Alternating planar algebras provide an appropriate algebraic framework for other knot invariants in cases the elements involved in the computation are alternating. The concept has been used in extending to tangles some properties of Jones polynomial and Khovanov homology of alternating links.

- ↑ Conway, J. H. (1970). "An Enumeration of Knots and Links, and Some of Their Algebraic Properties" (PDF). In Leech, J. (ed.).
*Computational Problems in Abstract Algebra*. Oxford, England: Pergamon Press. pp. 329–358. - ↑ Kauffman, Louis H.; Lambropoulou, Sofia (12 Jan 2004). "On the classification of rational tangles".
*Advances in Applied Mathematics*.**33**(2): 199–237. arXiv: math/0311499 . Bibcode:2003math.....11499K. - ↑ Ernst, C.; Sumners, D. W. (November 1990). "A calculus for rational tangles: applications to DNA recombination".
*Mathematical Proceedings of the Cambridge Philosophical Society*.**108**(3): 489–515. Bibcode:1990MPCPS.108..489E. doi:10.1017/s0305004100069383. ISSN 0305-0041.

- Adams, C. C. (2004).
*The Knot Book: An elementary introduction to the mathematical theory of knots*. Providence, RI: American Mathematical Society. pp. xiv+307. ISBN 0-8218-3678-1.

- MacKay, David. "Metapost code for drawing tangles and other pictures".
*Inference Group*. Retrieved 2018-04-13. - Goldman, Jay R.; Kauffman, Louis H. (1997). "Rational Tangles" (PDF).
*Advances in Applied Mathematics*.**18**: 300–332.

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